\[ \DeclareMathOperator{\div}{div} \DeclareMathOperator{\grad}{grad} \]

Let \(\Omega\subset \mathbb{R}^n\) be a domain . The Laplacian of a twice differentiable function \(f\colon \Omega\to \mathbb{R}\) is defined by

\begin{equation*} \Delta f := \div (\grad u) = \sum_{i=1}^{n} \frac{\partial^2 f}{\partial x_i^2}. \end{equation*}

Links Link to heading

• product rule
• Laplacian in polar coordinates

Equations Link to heading

• Laplace’s equation
• Poisson’s equation

Special Functions related to \(-\Delta\) Link to heading

• harmonic function
• eigenfunctions

Laplacians in other Spaces Link to heading

• Laplace-Beltrami
• Laplacian on \(\mathbb{Z}^d\)